multi-layer game-theoretical analysis of wireless markets ...elling wireless markets at a...

Multi-layer Game-Theoretical Analysis of WirelessMarkets with Market Segmentation

Georgios Fortetsanakis∗ Maria Papadopouli ∗†

Abstract New wireless access markets have emerged that are larger, more heterogeneous and diverse. Modelling suchmarkets can be challenging due to the interplay of various business- and network-related aspects as well as the interdepen-dencies among different entities (e.g., customers, providers). Existing models of wireless markets are either microscopic,focusing on a specific technical aspect (e.g., protocol, network topology, technology) at a fine scale or macroscopic mod-elling wireless markets at a large-scale, e.g., considering homogeneous user populations. In contrast to these approaches,this work develops a multi-layer game-theoretical framework, which allows providers to model users at multiple levels ofdetail by considering a different number of user sub-populations. It also models the mobility pattern, traffic demand, andnetworks of providers. A population game using Logit dynamics models the user selection of the appropriate dataplan andprovider, capturing the diversity in customer profile and relaxing the assumption about the user rationality. It analyticallycomputes the equilibriums of users and providers and numerically evaluates the performance of the market as a functionof the traffic demand, the number of available dataplans, and the knowledge about customer population. Significant ben-efits in revenue can be achieved by a provider when it integrates more detailed information about the user population.The number of disconnected users also decreases. Moreover the availability of several dataplans further enhances thegains. The stronger the provider, the more prominent the benefits. However the benefits diminish when all the providersmodel the customer population at the same degree of detail due to an increased competition. The analysis highlights thedevelopment of different strategies of the providers depending on their capacity, level of knowledge about the customerpopulation, and traffic conditions. It illustrates how a provider changes its strategy under different conditions, focusingpotentially on different customer segments and also the pressure introduced by specific customer types.

1 IntroductionThe wireless markets have experienced drastic changes and expansion, becoming more diverse (e.g.,in terms of customer profiles, services, providers), more complex and dynamic. The virtualization, in-creased mobile traffic, cloud infrastructures, and OTT applications have been changing the landscape.The telecom industry is at a crossroads, facing competitive challenges, such as the decline of legacyservices (e.g., voice and SMS have been rapidly supplanted by OTT applications), increased WiFiaccess, OTT competition, and saturation, that further enhances the competitive pricing pressure [2].Pricing can be used to regulate the user behaviour, manage the network resources, while optimizingthe Quality of Service (QoS) and revenue, especially as the demand may exceed the capacity. There-fore the design of appropriate pricing plans is important: high prices may result in high churn, whilelow prices may substantially increase the number of subscribers, which may result in congestion andcustomer dissatisfaction. A good balance between user requirements and profit can be achieved bythe identification of optimal plans. For that, various economic and technical aspects that affect the

∗University of Crete and Foundation for Research and Technology-Hellas, Greece.†Contact author: Maria Papadopouli ([email protected]).

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customer and provider decision making need to be modelled. Providers have applied various data-mining algorithms to identify the different market segments (e.g., [36, 15, 45, 21]). How does theknowledge about the different market segments impact the churn and the profit? How do providersdevelop their strategies and how do they change them to face the traffic increase and improve theirprofit? How does a weak provider “survive” in the presence of stronger providers? How do thedifferent market segments influence the market? Such questions can be answered using game-theory.

In general the game-theoretical modelling of wireless markets is performed either at a micro-scopic or at a macroscopic level: Microscopic approaches usually focus on specific technical as-pects (e.g., protocol, network topology, technology) on a short spatial and temporal scale, oftenconsidering networks with a small number of base stations (BSs) and markets with a single provider[11, 24, 44, 14, 46, 13, 48]. The computational and scalability issues when analysing large (e.g.,nation-wide) markets become prominent. On the other hand, macroscopic approaches model large-scale markets using homogeneous populations [17, 29, 30, 27, 16] and a simplified version of thenetwork infrastructure [43, 47] to make the analysis tractable. Such macroscopic approaches allevi-ate the computational issues but at the expense of accuracy, since they do not consider the diversityin the customer population. Unlike these approaches, this work develops a multi-layer two-stagegame-theoretical framework that allows providers to model users at multiple levels of detail basedon their profile. Our objective is twofold: to answer the aforementioned questions and address thecomputational-accuracy trade-off.

The competition of providers is modelled as a normal-form game, in which providers strategi-cally select their prices to optimize their revenue. The framework models in detail the network ofseveral providers, each offering multiple pricing plans to their customers. The plans are data-oriented(with prices based on data caps) and the focus is on wireless data traffic, as in most telecom marketscurrently. The framework captures important aspects of the customer behaviour, including their pref-erences and loyalty, traffic demand, mobility pattern (e.g., session arrival process, handovers), andQoS metrics. It identifies market segments (i.e., user sub-populations) based on their willingness-to-pay, datarate, and traffic demand, and model their decision making separately. Examples of suchchoice strategies consumers may use, especially when price is better known than quality, are the bestvalue, price-seeking, and price aversion [39]. A population game models the user decision-makingprocess. Via the Logit dynamics, a user decides to become subscriber of a certain provider or remaindisconnected.

Based on this modelling framework, we analysed the benefits of integrating information about themarket segments (e.g., consumer profiles) in different levels of detail (i.e., multi-layer aspect). Theevaluation focuses on the revenue of providers and percentage of disconnected users, as a functionof the level of detail, number of dataplans, and traffic demand. Significant benefits in revenue canbe achieved by a provider when it integrates more detailed information about the user population.The number of disconnected users also decreases. Moreover the availability of several dataplansfurther enhances the gains. The stronger the provider, the more prominent the benefits. Howeverthe benefits diminish when all the providers model the population at the same degree of detail, dueto an increased competition. The analysis highlights the development of different strategies of theproviders depending on their capacity, level of knowledge about the customer population, and trafficconditions. It illustrates how a provider changes its strategy under different conditions, focusingpotentially on different customer segments and also the pressure introduced by specific customertypes. It also shows how the weak provider reacts and “survives" the competition. With a largernumber of dataplans, providers can charge the different customer sub-populations more efficiently,achieving higher revenue.

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This paper extends our earlier work and the state-of-the-art in the following ways: (1) It enablesproviders to model users at different levels of detail and analytically computes the equilibriums ofusers and providers. (2) It employs a detailed model of the network. (3) It allows providers to offerseveral dataplans depending on user traffic demand and models the user-decision making, capturingimportant aspects of customer behaviour. (4) It relaxes the assumption about the user rationality inselecting providers and dataplans. The irrationality is often attributed to the subjective assessment onthe benefits and risks (as indicated by behavior economics).

The paper is structured as follows: Section 2 presents our modeling framework, focusing on thequeuing network of providers, user service selection, and the competition of providers. Section 3evaluates the performance of a wireless market when providers model users at different levels ofdetail. Section 4 discusses the related work. Finally, Section 5 presents our conclusions and futurework plan.

2 Modeling frameworkEconomic and technical aspects affect the decision-making process of customers and of providers.Our modelling framework consists of two layers, the technological layer and the economic one. Thetechnological layer models the wireless networks of providers as queueing networks and the usertraffic demand with appropriate stochastic processes. It also estimates the QoS of providers basedon the average and variance of data rate. The economic layer models the interaction among a setof providers and a heterogeneous user population. Each provider offers different dataplans to usersthat produce different traffic demand and selects their prices aiming to maximize its revenue, whileeach user selects a provider considering the offered prices and quality of service (QoS) guarantees.The user population is divided into groups, each having different characteristics and preferences.Throughout this paper, we use the terms users and customers interchangeably (similarly for the termsgroup and segment). Given the growth of LTE networks, the rapid increase of the WiFi mobile traffic(expected to consitute more than 46 percent of mobile data[6]), the evolution of the IP networks,and the decline of the voice and sms, we focus here on data (downlink) transmission and monthlydataplans.

A two-stage game defines the interaction of users and providers. The first stage instantiates thecompetition of providers and the second one the user-decision making. A population game modelsthe user decisions: members of each user group could either select to become subscribers of a certainprovider or remain disconnected. The decisions of these users are modeled by the Logit dynamics,a system of ordinary differential equations. They are based on a utility function that depends on theprice and QoS and a noise parameter that defines how much trust users place on this utility function,capturing the user “irrationality” and “stickiness” to a provider. On the other hand, the competitionof providers is modeled as a normal-form game in which providers strategically select their pricesto optimize their revenue. The utility functions of providers depend on the offered prices and theequilibrium of users (Fig. 1). Our framework models a wireless access market of I providers anda user population of J groups. Each provider has deployed a network of wireless BSs and offerslong-term subscriptions, which are best-effort data services. The following subsections describe thecomponents of our modeling framework in more detail.

3

…

…

…

…

……

……

Queueing network

Solve Traffic equations

(Eq. 3)

Estimate QoS metrics (average and variance

of data rate)

𝜌𝑖𝑘𝑗

𝐵𝑖𝑘

Technological layer

Equilibrium computation

Logit dynamics

User service selection

Provider utility functions

NE computation

Competition of providers

Prices (c)

𝑧∗(𝑐)

𝑐∗

Economic layer Two-stage game

Stage 2Stage 1

Gro

up

1 u

tilit

y

Gro

up

2 u

tilit

y

…

Gro

up

J u

tilit

y

User equilibrium estimation under different

degrees of knowledge

𝑧∗(𝑐)

𝜔𝑖𝑘𝑗

Figure 1: Overview of the main components of the framework: network of providers, market seg-mentation, traffic, customer selection, and pricing strategy of providers.

Table 1: Queueing-theoretical parameters for users of group j when connected at the networkof provider i

Parameter DescriptionKi Number of BSs of the provider iλ j Total session generation rate

z ji(z j0) Ratio of subscribers (disconnected users)ω

jik Steady-state probability for a user of group j to be located within the coverage of BS k

vik Departure rate from BS k due to handoverµik Session service rate at BS kdik Total departure rate from BS k (dik = vik + µik)

p( j)∗i,m,k Conditional prob. of handover from BS m to BS k given that a handover occurs

p( j)i,m,k Unconditional prob. of handover from BS m to BS k

(p( j)

i,m,k = vim p( j)∗i,m,k/dim

)γ

jik Total session arrival rate at BS k

a jik Arrival rate of new sessions at BS kρ

jik Traffic intensity at BS k

ni Vector indicating the number of users at each BSQi(ni) Stationary distribution of number of users at BSs

Bik Bandwidth at BS kR ji(z) Average data rateV ji(z) Variance of data rate

4

2.1 The queueing networks of providersEach provider (e.g., provider i) has deployed a number of BSs (Ki) covering a geographical region(e.g., a city). We assume that in all BSs, the available bandwidth is shared equally among connectedusers, i.e., processor-sharing discipline. 1. Users generate requests to connect to a base station (BS)to start a session. During a session, a user transmits and receives data via that BS. The sessiongeneration of a group of users j follows a Poisson process with a total rate of λ j. This rate is allocatedacross providers according to the probability vector z j = (z j0, z j1, ..., z jI). The ratio of members of thegroup j that select the provider i is indicated by z ji, while z j0 indicates the ratio of members of thatgroup that select the disconnection. The vector z = (z1, ..., zJ) corresponds to the probability vector ofeach user group and shows how the entire user population is divided among the available providersand disconnection state.

The mobility of members of the group j in the network of a provider is modeled with a Markov-chain in which a state corresponds to the coverage area of a BS. The total session generation rate ofthe members of the group j that select the provider i is further divided among its BSs (k = 1, ...,Ki)according to the probabilities ω j

ik. These probabilities correspond to the stationary distribution of theMarkov chain modeling the user mobility. Note that the handovers at a BS k of the provider i aremodeled with a Poisson process of total rate vik. This rate is estimated according to the fluid flowmobility model [34]. We also assume that handovers are performed in a seamless manner. Table 1defines the queueing-theoretical parameters for the members of the group j when connected at thenetwork of the provider i. Let us now focus on a simple case in which all users select the provideri (i.e., z ji = 1 for all j = 1, ..., J). The total session arrival rate at a BS k from members of thegroup j (γ j

ik) consists of the new sessions (a jik = ω

jikλ j) and handover sessions from neighbouring BSs

(Fig.2a):

γjik = a j

ik +

Ki∑m=1

γjim p( j)

i,m,k (1)

The traffic intensity generated by the users of the group j at the BS k of the provider i (ρ jik) is equal to

the ratio of the total session arrival rate at the BS k from members of the group j (γ jik) over the total

session departure rate at that BS (dik). However, to characterize the performance of the network of theprovider i, the total traffic intensity introduced by all user groups at each BS needs to be estimated.By summing the Eq. 1 over all user groups, we derive the traffic equations for the network of theprovider i:

J∑j=1

γjik =

J∑j=1

a jik +

Ki∑m=1

J∑j=1

γjim p( j)

i,m,k (2)

We now define the total session arrival rate at the BS k of the provider i from all user groups γik =∑Jj=1 γ

jik, the total arrival rate of new sessions at the BS k of the provider i aik =

∑Jj=1 a j

ik, and theaverage unconditional probability of a handover from the BS m to the BS k over all user groupspi,m,k =

∑Jj=1 γ

jim p( j)

i,m,k/γim. The corresponding average conditional probability of a handover from theBS m to the BS k given that a handover occurs is defined as p∗i,m,k = pi,m,kdim/vim Then, the trafficequations can be rewritten as in Eq. 3:

γik = aik +

Ki∑m=1

γim pi,m,k (3)

1Various scheduling algorithms that perform a long-term proportional fair channel allocation have been proposed inthe context of LTE networks [5, 19]Similarly, the IEEE 802.11 achieves a long-term fair bandwidth allocation.

5

The queueing network of the provider i is modeled as a Markov chain. Each state corresponds toa vector ni = (ni1, ..., niKi) indicating the number of connected users at all BSs. State transitionscorrespond to various types of events including session arrivals, terminations, and handovers. Thestationary distribution of the Markov chain is estimated by solving the global-balance equations.Such equations set the arrival rate at each state of the Markov chain equal to the departure rate fromthat state. Due to the Markovian property and the processor-sharing discipline of our system, theglobal-balance equations can be simplified into a set of local-balance equations [3]. Unlike global-balance equations, local-balance equations focus on the session arrivals and departures at specificBSs. According to these equations (Eqs. 4), the rate leaving a state ni due to the departure of a user ata specific BS k is equal to the rate entering that state due to the arrival of a user at the BS k either dueto a new session or a handover (Eq. 4a). Furthermore, the rate leaving the state ni due to the arrivalof a new session at a BS is equal to the rate entering that state due to the termination of a session at aBS (Eq. 4b).

dikQi(ni) = aikQi(ni − eik) +

Ki∑m=1

vim p∗i,m,kQi(ni − eik + eim) (4a)

Ki∑k=1

aikQi(ni) =

Ki∑k=1

µikQi(ni + eik) (4b)

In Eqs. 4, eik is a vector with all entries equal to 0 except the k-th entry which is equal to 1. Fig. 2billustrates the local-balance equations for a network of two BSs. Given that ρik =

∑Jj=1 ρ

jik < 1 for

each BS of the provider i, the stationary distribution of the number of connected users at all BSs canbe derived as follows:

Qi(ni) =

Ki∏k=1

(1 − ρik) (ρik)nik (5)

By substituting Eq. 5 in the local-balance equations (Eqs. 4) and using simple algebra, we derive thetraffic equations (Eq. 3). This proves the validity of Eq. 5. Given that the stationary distribution isin product form, each BS can be viewed as an independent M/M/1 queue with the processor-sharingdiscipline.

In the general case in which not all users select the provider i (i.e., z ji < 1), we can replaceγ

jik, a j

ik, and ρjik with z jiγ

jik, z jia

jik, and z jiρ

jik, respectively and Eqs. 1-5 still hold. In this case, the

average number of connected users at the BS k of the provider i is E[Nik] =ρik

1−ρik=

∑Jj=1 z jiρ

jik

1−∑J

j=1 z jiρjik

[23],

where ρik is the traffic intensity introduced by all user groups (ρik =∑J

j=1 z jiρjik). When a new user

arrives at the BS k, it shares the available bandwidth along with all other currently connected usersat that BS. Therefore, the amount of bandwidth that a new user gets when it connects to the BS k is

BikE[Nik]+1 = Bik(1 −

∑Jj=1 z jiρ

jik), where Bik is the total bandwidth of that BS. The average data rate as

perceived by a user of the group j at the network of the provider i can be computed as the weightedaverage of the data rate achieved at each BS (Eq. 6):

R ji(z) =

Ki∑k=1

ωjikBik(1 −

J∑l=1

zliρlik) (6)

6

BS kBS 𝑥 BS 𝑦

Handovers from BS 𝑦

𝛾𝑖𝑦𝑗𝑝𝑖,𝑦,𝑘(𝑗)

Handovers from BS 𝑥

𝛾𝑖𝑥𝑗𝑝𝑖,𝑥,𝑘(𝑗)

New

ses

sio

ns

Arr

ival

rat

e:𝑎𝑖𝑘𝑗

𝛾𝑖𝑘𝑗= 𝑎𝑖𝑘

𝑗+ 𝛾𝑖𝑥

𝑗𝑝𝑖,𝑥,𝑘(𝑗)

+ 𝛾𝑖𝑦𝑗𝑝𝑖,𝑦,𝑘(𝑗)

Users of group j

(a)

Term

inat

ion

at

BS 1

Arr

ival

at

BS 1

𝑛𝑖1𝑛𝑖2

𝑛𝑖1 − 1𝑛𝑖2

𝑛𝑖1 − 1𝑛𝑖2 + 1

𝜇𝑖1

𝑎𝑖1

𝜇𝑖1 + 𝑣𝑖1 𝑄𝑖 𝑛𝑖1, 𝑛𝑖2 =𝑎𝑖1𝑄𝑖 𝑛𝑖1 − 1, 𝑛𝑖2 + 𝑣𝑖2𝑄𝑖 𝑛𝑖1 − 1, 𝑛𝑖2 + 1

𝑛𝑖1𝑛𝑖2

𝑛𝑖1𝑛𝑖2 + 1

𝑛𝑖1 + 1𝑛𝑖2

𝜇𝑖1

𝑎𝑖1

Term

inat

ion

at

BS

1

Arr

ival

at

BS 1

𝑎𝑖1 + 𝑎𝑖2 𝑄𝑖 𝑛𝑖1, 𝑛𝑖2 =𝜇𝑖1𝑄𝑖 𝑛𝑖1 + 1, 𝑛𝑖2 + 𝜇𝑖2𝑄𝑖 𝑛𝑖1, 𝑛𝑖2 + 1

State of the Markov chain𝑛𝑖1, 𝑛𝑖2: Num. of users at BSs 1 & 2

Transition rate from state 𝑛𝑖1, 𝑛𝑖2 + 1 to state 𝑛𝑖1, 𝑛𝑖2

Eq. 4a for BS 1 Eq. 4b

(b)

Figure 2: Session arrivals at a BS of the provider i from members of the group j (left). The local-balance equations for a network with two BSs (right).

The spatial variability of data rate affects the QoS. Thus, the utility function of the members of thegroup j (Eq. 8) incorporates the average data rate (Eq. 6) and variance of data rate which is definedas a polynomial of second degree with respect to z (Eq. 7):

V ji(z) =

Ki∑k=1

ωjik

Bik(1 −J∑

l=1

zliρlik) − R ji(z)

2

(7)

The user service selection process employs the average and variance of data rate. The sub-gamesmodeling the user service selection and competition of providers are described in Subsections 2.2and 2.3, respectively.

2.2 User service selectionThe user service selection process is modeled by a population game. Each member of a user groupcan choose among I + 1 available strategies H = {0, 1, ..., I}. Strategies 1, 2, ..., I correspond tosubscriptions with the providers 1, 2, ..., I, respectively, while strategy 0 denotes the disconnectionstate. We assume that each group corresponds to a homogeneous sub-population, and as such, theutility attained when selecting a specific strategy is the same for all users in that group. Therefore, itsuffices to describe the service selection of the members of the group j with a probability distributionover the set of strategies (H). This distribution z j = (z j0, z j1, ..., z jI) is the strategy profile of the groupj indicating how members of this group are divided among the different strategies (subscriptions anddisconnection). The strategy profile of the entire user population consists of the strategy profiles ofall groups (z = (z1, ..., zJ)). Additionally, the market share that corresponds to a strategy i is theaverage percentage of customers over all user groups that select this strategy, i.e., zi =

∑Jj=1 z jiN j/N.

All parameters of a wireless market are defined in Table 2.Utility function of the group j. A user from the group j selects a strategy (i.e., a subscription or

disconnection) based on the average and spatial variance of the achievable data rate at the networks

7

Table 2: Main parameters of a wireless market

Parameter DescriptionI Number of providersJ Number of user groups

N j Number of users in group jN Total number of usersc Vector with the prices of all providersH User strategies

f j

(R ji(z)

)Impact of average data rate on utility function of group j

w jV (w j

P) Weight of variance of data rate (price) for users of group ju ji (z; c) Utility function of group j

d j Average monthly traffic demand in MB of a member of group jz(t) User strategy profile at time t

z∗(c) Equilibrium of users when price vector is cP ProvidersC Provider strategy profiles

σi(c) Utility function of provider i

of providers and the offered prices:

u ji(z; c) =

f j

(R ji(z)

)− w j

VV ji(z) − w jPci(d j) if i = 1, ..., I

0 if i = 0(8)

The function f j is concave, strictly increasing, and non-negative and defines the impact of the averagedata rate (R ji(z)) on the utility of the members of the group j. Such functions have been frequentlyused in the literature (e.g., [12, 20]). In the analysis, we assume that f j is exponential, i.e., f j(x) =

w jR(τ j − exp(−h jx)). Its main characteristic is that it defines a diminishing return for users when the

QoS improves2. The parameter w jR expresses the willingness to pay of the members of the group j.

The larger the w jR, the larger the maximum price that users from the group j can pay. The parameter

h j defines the sensitivity of the members of the group j to low data rate. The larger the h j, the largerthe tolerance of users to a low data rate. The impact of the variance of data rate (V ji(z)) and price ofthe subscription of the provider i (ci(d j)) is assumed to be linear and their significance is indicated bythe positive weights w j

V and w jP, respectively. The price that the members of the group j pay when

selecting the subscription with the provider i depends on the average traffic demand they produce ina period of a month (d j) according to Eq. 9:

ci(d j) =

ci1 if 0 < d j ≤ D1

ci2 if D1 < d j ≤ D2

...

ciS i if DS i−1 < d j ≤ DS i

(9)

Dataplans. Most of the operators employ pricing strategies based on the data instead of talk timeor text, in response to the changes in the usage patterns (e.g., [31]). The provider i offers S i distinct

2Other utility functions, e.g., logarithmic and isoelastic ones, could be also employed.

8

dataplans each for different traffic demand levels. Depending on which interval the traffic demand ofa user lies, that user pays a different price. In other words, the provider i charges each user with aflat rate that depends on its level of traffic demand according to Eq. 9. We assume that each user isaware of its own average traffic demand per month when selecting a service 3. Furthermore, when auser selects the disconnection (i.e., i = 0), it attains utility equal to 0. The parameters of the utilityfunction u ji (i.e., w j

R, τ j, h j, w jV , w j

P and d j) constitute the profile of the members of the group j.Evolution of user-decision making. Based on the utility functions of all user groups, the evolu-

tion of the strategy profile of users (z(t)) is described by the Logit dynamics, a system of ordinarydifferential equations (Eq. 10). Compared to other population dynamics, the Logit dynamics havetwo attractive properties: (1) They can capture the user bounded rationality and stickiness/loyalty tocertain providers. (2) They are innovative dynamics.

dz ji(t)dt

=r

1 +∑

k,i exp(u jk(z(t))−u ji(z(t))

ε

) − rz ji(t) (10)

Note that we assume that the evolution of the user decisions manifests at a much slower time scalecompared to the time scale at which sessions arrive and depart at the BSs of a provider. Specifically,the session arrivals and departures are performed at a time scale of minutes, while the user decisionsmanifest at a time scale of days or even months. Therefore, when the user strategy profile changesfrom z to z0 over the course of several days, there is enough time for the queueing network of aprovider to reach the equilibrium. We can then assume that the average data rate changes from R ji(z)to R ji(z0) and the spatial variance of data rate changes from V ji(z) to V ji(z0). The parameter r controlsthe speed of the dynamics, while ε is the noise and takes values in the interval [0,∞). When ε = 0,users are completely “rational”, i.e., always select the strategy that maximizes their utility function.Specifically, if the strategy k has slightly larger utility than strategy i that has been selected by arational user, this rational user will then switch to strategy k. However, this behaviour is not realistic.

Users are reluctant to switch to another provider when the additional benefits of switching aresmall. According to a survey study, users should be offered additional benefits of around 40% beforethey are highly likely to change their provider [42]. Several aspects, such as, the brand name andbrand equity [38], reputation of a provider [22, 8], market share of a provider [18], length of customerassociation with a provider [37], and force of habit [25], affect the user-decision making. In this work,the noise of the Logit dynamics models the aggregate effect of those aspects. As the noise increasesfrom 0 towards infinity, users become “stickier” with their selected service and change provider onlywhen the benefits in terms of price and QoS are large enough.

To compute the equilibrium of users for a given set of prices offered by the providers, we solvethe system of the Logit dynamics (Eq. 10) using a standard ODE solver starting from an initial pointat which all user groups are uniformly distributed across the available strategies (i.e., subscriptionswith providers and disconnection). The point at which the Logit dynamics converge is the userequilibrium.

2.3 Competition of providersThe competition of providers is modelled as a normal-form game (P,C, {σi}i∈P). In this game, eachprovider (say provider i) offers S i distinct dataplans to users. The strategy of that provider ci =

3In this paper, the term ”service" refers to a subscription with a provider or the disconnection, while the term ”data-plan" refers to the offered price of a provider for a specific interval of traffic demand. A user selects a service and paysthe price corresponding to its level of traffic demand.

9

0 10 20 30 40 500

0.5

1

1.5

2x 10

6

Price of provider 1

Utilit

y fu

nctio

n of

pro

vider

1

(a)

0 10 20 30 40 500

0.5

1

1.5

2x 10

6

Price of provider 1

Utilit

y fu

nctio

n of

pro

vider

1

(b)

0 10 20 30 40 500

0.5

1

1.5

2x 10

6

Price of provider 1

Utilit

y fu

nctio

n of

pro

vider

1

(c)

Figure 3: Effect of noise ε on the utility function of a provider in a market with 5 user groups, ε = 0(left), ε = 0.5 (middle), and ε = 1.5 (right).

(ci1, ..., ciS i) is a vector containing the prices of all those dataplans. Each price is selected from aclosed interval [0, Cmax

i ]. The strategy space of providers is the set of all possible combinations ofprices that can be offered in the market and is a rectangle of the form C = [0, Cmax

1 ]S 1 × [0, Cmax2 ]S 2 ×

... × [0, CmaxI ]S I . Each point of the strategy space c = (c1, ..., cI) is a vector containing the offered

prices of all providers.We assume that each provider can model the user population at different levels of detail. They

analyze customer data, e.g., demographic information as well as information about their traffic de-mand and profile, to perform market segmentation. In the performance evaluation, we considered asynthetic user profiles dataset (for the entire customer population) and that each provider can extractinformation for a different number of market segments (user clusters), corresponding to an incom-plete view of the market. 4 The larger the number of market segments, the more detailed informationabout the user population.

Each provider, after estimating the market segments of users, it models their decision makingbased on the system of Logit dynamics (Eq. 10). For each set of offered prices c, a provider solvesthe system of Logit dynamics modeling the decision making of these clusters of users starting fromuniform initial conditions and computes the corresponding equilibrium point (z∗(c)). Based on thisequilibrium, the utility function of the provider i is defined according to Eq. 11:

σi(c) =

S i∑s=1

∑j:Ds−1<d j≤Ds

N jz∗ji(c)cis (11)

This function computes the sum of the revenue collected by each of the offered dataplans of theprovider i. For a dataplan s, only the payments of the clusters corresponding to this dataplan are con-sidered (i.e., the user clusters that produce an average traffic demand lying in the interval (Ds−1,Ds]).

When a user is rational (i.e., ε = 0), it selects the provider that maximizes its utility function. Letus assume that such a user has selected the provider i. If at a certain point the subscription of anotherprovider k becomes slightly more profitable compared to the subscription of the provider i, the userwill switch providers. This extreme behaviour of rational users introduces technical difficulties in theanalysis: Various discontinuities appear in the derivatives of the utility functions of providers. Wedeveloped a methodology to analytically compute the NEs of providers under such discontinuities.This methodology divides the strategy space of providers into various subsets called “regions” in

4Obviously the analysis can be extended considering that providers have access to different datasets of the customerpopulation.

10

which the provider utility functions are continuously differentiable and analyses the game of providersat each one of those regions separately. At the final step, it combines the results of the analysis from allthe regions to compute the global NEs of providers. A more detailed description of this methodologycan be found in our earlier work [9].

This modeling approach can effectively compute the NEs of providers, when users are completelyrational and providers model users at the macroscopic level (i.e., considering only 1 cluster) and offeronly 1 dataplan5. However, what happens when providers have a more rigorous knowledge about themarket segments? In such a case, the computation of the global NE becomes challenging: As thenumber of user clusters increases, the number of (hyper-)surfaces at which the utility functions ofproviders have discontinuous derivatives in the strategy space of providers also increases.

This issue is mainly due to the assumption of the user rationality. Rationality is a set of principlesthat describes the utility maximinzing choice. Under this assumption, users can drastically changetheir behaviour even under small deviations in their utility function. To more realistically model theuser behaviour, we use the Logit dynamics (Eq. 10). Among other standard population dynamics,such as, the replicator, best response, BNN, and Smith dynamics, these are the only ones that canmodel the user irrationality [35]. As explained earlier, their noise parameter (ε) indicates how muchthe user decisions deviate from the optimal ones based on their utility function capturing the effectof other factors that influence users (e.g., psychological/social factors). In general, issues such as theprice-quality correlation, impact of objective information and personal experience on user strategyhave been examined in marketing e.g., [39].

The noise of the Logit dynamics does not only make the model of users more realistic but alsosimplifies the analysis by “smoothing out” the discontinuities in the derivatives of the utility func-tions of providers. An example of the effect of noise is shown in Fig. 3. This figure presents theutility function of a provider (say provider 1) offering only 1 dataplan in an oligopoly with 5 distinctuser groups, when the prices of its competitors remain fixed. When the noise is equal to zero (i.e.,ε = 0), the rational behaviour of users results in various discontinuities in the derivative of the utilityfunction of the provider 1 (Fig. 3a). When the value of noise slightly increases (i.e., ε = 0.5), the dis-continuities are smoothed out (Fig. 3b). If the noise increases a little further (i.e., ε = 1.5), the utilityfunction of the provider becomes concave. This function (Fig. 3c) does not deviate significantly fromthe corresponding one when users are completely rational (Fig. 3a).

In each market case, by selecting a sufficient amount of noise, the utility functions of providersbecome concave. This simplifies the estimation of a global NE significantly. One should simply setthe derivatives of the utility functions of providers with respect to their prices equal to 0:

∂σi(c)∂cis

= 0, for all i = 1, . . . , I and s = 1, . . . , S i (12)

To compute the Nash equilibrium of providers, we solve the system of non-linear Eqs. 12 usinga standard numerical-analysis algorithm. When the utility functions of providers are concave, asolution of the system 12 (c∗) is guaranteed to be a global NE of the game of providers. At thispoint, the utility functions of providers are maximized given the prices of their competitors [4] andtherefore, no provider has the incentive to change its strategy. Our algorithm for the estimation ofa NE proceeds as follows: First the system of Eqs. 12 is solved. If a solution is reported, it willbe then verified whether or not it corresponds to a global NE (i.e., at which the utility function ofeach provider is maximized given the prices of the other providers). This final verification step is

5In the worst case, the algorithm needs to solve two non-linear systems of I equations and a system of 2I non-linearinequalities.

11

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(b), respectively. Distribution of user groups for independent w jR & h j (c).

necessary given that the concavity of the utility functions of providers is not always guaranteed. Forexample, if the selected value of noise is not sufficiently large, the utility function of one or moreproviders may not be concave. In such a case, for a point to be a Nash equilibrium, the system ofEqs. 12 is still necessary but not sufficient. Therefore, a solution of Eqs. 12 could correspond to alocal maximum of the utility functions of providers, i.e., a “local NE” instead of a global one.

3 Performance evaluationWe implemented the modeling framework in Matlab and instantiated a wireless access market of asmall city, represented by a rectangle of 14.4 km x 12.5 km. This market includes 4 providers anda population of 300, 000 users. Each provider has deployed a cellular network covering the entirecity. The BSs at each network are placed on the sites of a triangular grid, with a distance betweentwo neighbouring sites of 1.6 km. The maximum data rate with which a BS can serve sessions is 25,22, 19, and 16 Mbps for the providers 1, 2, 3, and 4, respectively. The provider 1 is the strongestprovider in the market with the largest capacity at its BSs (i.e., 25 Mbps), while the provider 4 isthe weakest one with the lowest capacity (i.e., 16 Mbps). The average size of a session is 10 MB.Furthermore, the session service rate of a BS is µ1 = 18.75, µ2 = 16.50, µ3 = 14.25, and µ4 = 12.00sessions/min for the providers 1, 2, 3, and 4, respectively. We also consider different cases for theuser traffic demand (i.e., an average user session generation rate from 0 up to 1.5 sessions/hour) andanalyse its impact on providers and users.

3.1 Modeling users at different levels of detailWe consider a heterogeneous user population consisting of 100 distinct groups, which represents themost detailed picture of the user population. To model a diverse user population, we selected themaximum willingness to pay (w j

R) and tolerance on low data rate (h j) of these groups follow normaldistributions of mean 30 and 0.6 and standard deviation of 7.6 and 0.3, respectively. In real markets,those two parameters are often correlated. For example, users with a large willingness to pay (highw j

R) usually demand a high QoS and are less tolerant on low data rate (low h j). This is due to thefact that the service charge creates an expectation for the perceived quality. However to highlight theimpact of the correlation between the user willingness to pay and tolerance on low data rate, we alsodefined a scenario in which those two parameters are independent. Specifically, in the first (main)

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Figure 5: Performance gains when providers model users at different levels of detail compared tomacroscopic modeling when w j

R and h j are correlated (top) and when they are independent (bottom),respectively.

scenario, the cross correlation of w jR and h j is equal to −0.85 (Fig. 4a). This means that users with a

large willingness to pay (high w jR) usually are less tolerant on low data rate (low h j). In the second

scenario, the cross correlation of w jR and h j is equal to 0 and the maximum willingness to pay and

data rate requirements of groups are completely independent (Fig. 4b). In both scenarios, the trafficdemand (d j) is the same for all user groups and providers offer only one dataplan. Additionally, theweight of data rate variability (w j

V) is set equal to 0 for all user groups, while the noise value (ε)is set equal to 1.5. The noise parameter should reflect the user behaviour. However given the lackof real-world datasets, the minimum value of noise that results in concave utility functions (for theproviders) was selected.

We distinguish four distinct user categories the business users, low profile users, value for-moneyusers, and lenient users. Business users have a high willingness to pay (i.e., high value of w j

R) butare highly sensitive on low data rate (i.e., low value of h j). Low-profile (or basic) users are theopposite: They can not afford a high price but are more tolerant on low data rates. Value-for-money(or best-deal) users are the most demanding ones in the market. They have a low willingness topay and can not tolerate low values of data rate. Finally, lenient users are characterized by a highwillingness to pay and high tolerance in low data rate. These users make their decisions consideringmainly the price, i.e., they search for the cheapest service. Such consumer classification in terms ofchoice strategies is common in marketing, especially when price is better known than quaility [39].We clustered the 100 sub-populations to the above four categories. The characteristics of these fourcategories are given in Table 3.

For each scenario, we have defined different market cases in which providers model the user

13

Table 3: The four main user categories

Category Willingness to Tolerance on lowpay (w j

R) data rate (h j)Business users > 50% percentile < 50% percentile

Low-profile users < 50% percentile > 50% percentileValue-for-money users < 50% percentile < 50% percentile

Lenient users > 50% percentile > 50% percentile

population at different degrees of detail when estimating their utility functions. Each provider appliesa clustering algorithm (e.g., K-means) on the profiles of the 100 user groups and determines a setof 5, 9, or 20 representative user clusters. Providers consider these sets of clusters when estimatingtheir utility functions (according to Eq. 11). As the number of clusters increases, the modeling ofthe user population becomes more accurate but the computational complexity of estimating the NEof providers increases. Our objective is to select the most appropriate level of detail that results inlarge performance gains for providers at a low computational complexity. Fig. 5 shows the additionalbenefits of users and providers obtained when providers model the user population at different levelsof detail compared to modeling users macroscopically (i.e., with only 1 cluster) when w j

R and h j arecorrelated (Figs. 5a - 5c) and when they are independent (Figs. 5d - 5f), respectively.

Case a: When the maximum willingness to pay (w jR) and tolerance on low data rate (h j) are

correlated, modeling the user population at a higher level of detail (with a larger number of clusters)pays off for providers and users. Providers increase their revenue in almost all cases (Figs. 5band 5c) and the percentage of disconnected users is significantly reduced (Fig. 5a). When providersmodel users at the macroscopic level, they apply a homogeneous marketing strategy by selecting theirprices considering only aggregate profiling information for the entire population, without detailedinformation for the different sub-populations that reflect the diversity of the market. The provider1 offers a price that is close to the prices of the other providers and attracts both business and low-profile users. Additionally, the providers 3 and 4 offer relatively high prices losing customers andrevenue form the low-profile user groups. In such a market, except from the revenue of providers, theperformance of users is also sub-optimal. A large percentage of users ends up selecting the provider1 resulting in a degradation of the offered quality of service. Furthermore, due to the relatively highprices of the providers 3 and 4, more low-profile users become disconnected. In other words, under amacroscopic view of the market, providers make suboptimal decisions, which have a negative impacton their revenue and performance of users.

By employing a larger number of clusters for the estimation of their utility functions (e.g., 5, 9 or20 clusters), providers obtain a more detailed picture of the market, and as a result, the performanceof the market improves. The provider 1 offers a higher price compared to the other providers and at-tracts mostly business users, while its share of low-profile users drops. On the contrary, the providers3 and 4 offer low prices and attract the largest percentage of low-profile users, while their share ofbusiness users becomes low. The strong providers (i.e., the ones with the largest cellular capacity)focus on users with high willingness to pay and QoS requirements, while the weak providers focuson users with low willingness to pay and QoS requirements. This improves the overall performanceof users (Fig. 5a) and reduces the intensity of competition allowing for a higher revenue for providers(Figs. 5b and 5c). In general, as the number of clusters increases, the performance of the market im-proves. However, the computational complexity of computing the equilibriums of users and providers

14

Figure 6: Execution time needed to perform the market analysis and additional revenue gain of aprovider at different levels of detail.

increases (e.g., a non-linear increase in the execution time as shown in Fig. 6)6.Under high traffic demand, a large decrease of the benefits of providers is observed from a certain

point onwards (Figs. 5b and 5c). At this point, the capacity of the networks of providers is reachedand disconnected users start appearing. The first users that become disconnected are the ones with alow willingness to pay. Providers enter a competition and restrain their prices aiming to prevent thoseusers from becoming disconnected. Even the provider 1 enters the competition regardless of its focuson business users. As the traffic demand keeps increasing, it becomes less beneficial for providers tokeep those users in the market. This weakens the competition allowing for a small recovery of therevenue gains of providers.

Case b: If the maximum willingness to pay (w jR) and tolerance on low data rate (h j) are inde-

pendent, the modeling of users at a finer level of detail is beneficial for users: The percentage ofdisconnected users is significantly reduced (Fig. 5d). Interestingly, the performance of providers isnot always improved. Under a low traffic demand, providers achieve significant revenue gains, whileunder large traffic demand, they lose revenue compared to a macroscopic modeling of users (Figs.5e and 5f). Furthermore, the larger the number of clusters, the more prominent the losses. This isa counter-intuitive result: One would expect that the larger the degree of knowledge about the userpopulation, the more significant the benefits of the providers. To explain this phenomenon, we shouldfocus on the distribution of the user groups in this scenario (shown in Fig. 4c).

We distinguish four different types of groups: the business users, low-profile users, lenient users,and value-for-money users. The revenue losses are mainly due to the pressure that value-for-moneyusers and lenient users introduce in the market. Value-for-money users have a low willingness to pay(i.e., low w j

R) and small tolerance on low data rate (i.e., low h j). Satisfying the requirements of thoseusers can be difficult. Providers should offer services of high data rate on low prices. Under a largetraffic demand, when the capacity of the networks of providers is reached, value-for-money usersstart becoming disconnected. When modeling users at high levels of detail, providers are aware of

6It is in the interest of a provider to select the appropriate number of clusters that results in high revenue and requiresa relatively low execution time, if there are time constraints (e.g., 9 clusters).

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Figure 7: Performance gains of a market in which only the provider 1 models users with 9 clusters,while all other providers model users macroscopically compared to a market in which all providersmodel users macroscopically. The top (bottom) figures correspond to a scenario in which w j

R and h j

are correlated (independent), respectively.

the presence of value-for-money users and restrict their prices in an effort to attract these users giventhat they correspond to a significant percentage of the market (around 25%). This results in a steepdecrease of the provider revenues (Figs. 5e and 5f). As the number of clusters increases, providersbecome aware of more “extreme” cases of value-for-money users with stricter price and data raterequirements. Therefore, they become more aggressive in the decrease of their prices, and as a result,they lose more revenue.

In the case of high traffic demand, value-for-money users eventually become disconnected andtheir influence weakens allowing for a slow recovery of the provider revenues. However, in the caseof the provider 4, its revenue gain always remains negative under large traffic demand (Fig. 5f). Thisis due to the influence of lenient users. These users have a high willingness to pay (high w j

R) and hightolerance on low data rate (high h j). Price is the parameter that mainly drives their decisions. Undera large traffic demand, disconnected users appear in the value-for-money, low-profile and businessusers. However, almost all lenient users remain connected due to their low data rate requirements andhigh willingness to pay. Therefore, as the traffic demand increases, the influence of lenient users inthe market intensifies. This strengthens the competition of providers keeping their revenues low.

Impact of different degrees of detail in the knowledge about customers among providers. Werepeated the analysis, considering now a market in which only the provider 1 models users with9 clusters, while all other providers model users macroscopically. Figs. 7a, 7b, and 7c show thedifferences in the prices, market share, and revenue of providers, respectively, compared to a marketin which all providers model users macroscopically when w j

R and h j are correlated. Figs. 7d, 7e, and

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Figure 8: Performance gains of a market in which only the provider 4 models users with 9 clusters,while all other providers model users macroscopically compared to a market in which all providersmodel users macroscopically. The top (bottom) figures correspond to a scenario in which w j

R and h j

are correlated (independent), respectively.

7f present the same differences when w jR and h j are independent.

The provider 1 always achieves revenue gains. This is observed both when w jR and h j are cor-

related and when they are independent (Figs. 7c and 7f, respectively). With its higher degree ofknowledge, the provider 1 can influence the market to its benefit outsmarting the other providers.This is an important result which shows that the benefits of a provider from modeling the user popu-lation in a high level of detail strongly depend on the level of knowledge of the other providers aboutusers.

Another interesting trend is that the effect of the knowledge of the provider 1 on the revenues ofthe other providers is twofold. Under small traffic demand, the provider 1 raises its price comparedto the one offered at the macroscopic level (Figs. 7a and 7d). Given that it is the most influentialprovider in the market, its price increase provides also an opportunity to the other providers to raisetheir prices. This results in significant revenue gains for all providers (Figs. 7c and 7f). However,when the traffic demand becomes large, suddenly, the provider 1 “turns against” the other providers.When the capacity of the networks of providers is reached and disconnected users appear, it reducesits price bellow the prices of the other providers to attract those users (Figs. 7a and 7d). Given theirmacroscopic view of the market, the other providers do not react. This results in an increase of therevenue gain of the provider 1 at the expense of the revenues of the other providers (Figs. 7c and 7f).

Let us now explain the behaviour of the provider 1 in more detail. When the user traffic demandis low, the provider 1 realizes that it is more beneficial to focus on business users increasing its price.

17

This results in a decrease of its market share compared to the one obtained at the macroscopic level(Figs. 7b and 7e). However, this decrease of market share is useful because it improves the qualityof service of the provider 1 making its subscription more appealing to business users. Given theirimproved satisfaction, those users can pay a higher price to the provider 1 increasing its revenue.This trend persists as the traffic demand increases until the capacity of the networks of providersis reached. From this point onwards, the quality of service drops substantially and some of thebusiness users become dissatisfied and decide to disconnect. Then, the provider 1 suddenly changesits strategy. Instead of focusing on the business users, it realizes that it is more beneficial to restrict itsprice in order to prevent users with low willingness to pay from becoming disconnected and attractthem to its network. Other providers do not realize this early enough due to their macroscopic viewof the market and they do not react. That way, the provider 1 achieves significant revenue gain at theexpense of the revenues of the other providers.

We repeated the previous analysis considering now that only the provider 4 models users with9 clusters, while all other providers model users macroscopically (Fig. 8). The provider 4 achievesrevenue gains both when w j

R and h j are correlated (Fig. 8c) and when they are not (Fig. 8f). Addi-tionally, when the traffic demand is low, all providers achieve revenue gains, while under large trafficonly the provider 4 gains additional revenue at the expense of the revenues of the other providers(Figs. 8c and 8f). However, given that the provider 4 is the weakest one in the market, its influence islow, and therefore, the observed tends of Fig. 8 are subtler compared to Fig. 7.

3.2 Offering of multiple dataplansWe have defined a market in which users of different groups deviate not only in their willingness topay (w j

R) and tolerance on low data rate (h j) but also in their traffic demand (d j). As in Section 3.1,w j

R and h j follow normal distributions, while the average session generation rate over all user groups(λ =

∑Jj=1 λ j/

∑Jj=1 N j) varies from 0 up to 1.5 sessions/hour. The normalized session generation rate

n j =λ j/N j

λof user groups follows a normal distribution of mean 1 and standard deviation 0.2. Some

user groups correspond to “heavy” users producing a large amount of traffic (users groups with alarge n j), while other groups correspond to “light” users (groups with a low n j). Again we distinguishtwo market scenarios. In the first scenario, the cross correlation of w j

R and h j is equal to −0.85, whilethe cross correlation of w j

R and n j is 0.85. This means that users with a large willingness to pay(high value of w j

R) usually are heavy users (high value of n j) and less tolerant on low data rate (lowvalue of h j). In the second scenario, the cross correlation among w j

R, h j, and n j is equal to 0 andthe maximum willingness to pay, data rate requirements and traffic demand of groups are completelyindependent. In this set of experiments, we set the noise parameter of the Logit dynamics modelingthe user-decision making (ε) equal to 2.

We assume that a provider i offers a total number of S i dataplans each corresponding to a differentinterval of traffic demand and flat-rate price. The first dataplan is ”addressed" to light users with anormalized session generation rate up to 100/S i% percentile, while the last dataplan is appropriate toheavy users with normalized session generation rate lying in the interval from (S i − 1) ∗ 100/S i% upto 100% percentile. In other words, the range of the values of the normalized user session generationrate is divided into S i segments. Users groups belonging to different segments are charged with adifferent price. Providers can also model users at different levels of detail by estimating a differentnumber of user clusters. We have analysed a market in which providers offer 3 dataplans to users.The performance gains obtained when providers model users at different levels of detail compared to

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R, h j, and n j tobe correlated (top) and independent (bottom), respectively.

when they model users macroscopically and offer only 1 dataplan are presented in Fig. 9. The top(bottom) figures correspond to the case that w j

R, h j, and n j are correlated (independent), respectively.Similar trends are observed as in Fig. 5. When w j

R, h j, and n j are correlated, in most cases,providers achieve revenue benefits when they model users at a higher level of detail (Figs. 9b and9c), while the reduction of the percentage of disconnected users becomes more prominent (Fig. 9a).However, in a small interval around 0.95 session/hour, providers lose a small amount of revenue whenthey model users in higher detail. When the number of user clusters increases, an increase in thenumber of dataplans is required for a better pricing. With a larger number of dataplans, providers cancharge the different user clusters more efficiently achieving higher revenue. In the cases of 9 clustersand 20 clusters, when the number of dataplans is increased above 3, the revenue losses around 0.95sessions/hour are reduced. We omit those results due to lack of space.

When w jR, h j, and n j are independent, the observed trends are exactly the same as in Fig. 5. An

increase of the number of user clusters results in a more prominent reduction of the percentage ofdisconnected users (Fig. 9d). Additionally, under a low traffic demand, an increase of the numberof clusters always results in revenue benefits for providers. However, under a large traffic demand,modeling users with a large number of clusters results in revenue losses compared to the macroscopiccase (Figs. 9e and 9f). Again those revenue losses are due to the existence of value-for-money usersand lenient users. Those users intensify the competition of providers under a large traffic demandresulting in lower offered prices and revenue.

We have also studied a market in which providers offer a different number of dataplans (i.e., 1,3, or 5 dataplans). In this market, each provider models the users with 9 clusters. The performance

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gains compared to a market in which providers model users macroscopically and offer only 1 dataplanare depicted in Fig. 10. In the top (bottom) figures the parameters w j

R, h j, and n j are correlated(independent), respectively. When w j

R, h j, and n j are correlated, if providers offer only 1 dataplan,at an interval of the traffic demand around 0.9 sessions/hour, they achieve revenue losses comparedto the macroscopic case (Figs. 10b and 10c). Those losses are mainly due to the limited number ofdegrees of freedom when setting the prices of providers. Specifically, providers offer the same priceboth to heavy users with a large willingness to pay and to light users with a low willingness to pay. Toprevent light users from becoming disconnected, providers restrict their prices regardless of the highwillingness to pay of heavy users losing revenue. However, when providers offer 3 or 5 dataplans,the heavy and light users are charged with a different price. Therefore, providers are able to offer ahigh price to heavy users and a low price to light users always achieving revenue benefits (Figs. 10band 10c).

When w jR, h j, and n j are independent, a counter intuitive result is observed. An increase in the

number of offered dataplans does not result in revenue benefits for providers. On the contrary, undera large traffic demand, the offering of a larger number of dataplans results in revenue losses (Figs.10e and 10f). In this case, the willingness to pay and traffic demand of user clusters are independent.This means that the average willingness to pay of heavy and light users is almost the same. Whenproviders offer different dataplans to these users, their competition is intensified. Given that thelight users produce a low amount of traffic, their admission at the network of a provider does notsignificantly affect its QoS. Therefore, providers have the incentive to reduce their offered prices tolight users in order to attract them to their networks entering a price war. Additionally, providers cannot charge the heavy users with a high price due to their relatively low willingness to pay (almost thesame as the one of light users). Therefore, providers would lose revenue by offering a larger numberof dataplans in this market.

4 Related workThe game-theoretical approaches in analysing wireless markets [40, 28] can be classified into twogeneral categories, namely, the microscopic approaches and the macroscopic ones. Microscopicmodels usually focus on a short spatial and temporal scale and evaluate the impact of various technicalaspects (e.g., mobile data offloading [13, 48], femtocells [26], network selection mechanisms forusers [11, 1], cooperative spectrum access schemes for primary and secondary users in cognitiveradio networks [24, 41, 44], and multihop access paradigms [7]) on the performance of a network,providers, and users satisfaction. These modeling approaches can be detailed and accurate but usuallynot scalable or amenable to theoretical analysis. On the other hand, macroscopic approaches focuson larger-scale phenomena and make various simplifications [17, 29, 30, 27, 16]. For example, theymay consider a homogeneous user population and model the aggregate user behaviour (e.g., trafficdemand, QoS preferences [33, 32]), trading the accuracy for scalability and tractability.

To the best of our knowledge, the proposed framework is one of the few game-theoretical ap-proaches that models the users, network, and providers in detail. For example, it models relativelylarge-scale networks (at the BS level), the user demand, profile (e.g., sensitivity to the price and QoS,rationality and loyalty to certain providers), handovers, and mobility pattern. In our earlier work, wedeveloped an event-based simulator of a wireless access duopoly that could be executed at multiplelevels of detail from the microscopic to the macroscopic one [10]. However, no analytical resultswere derived for the user and provider equilibriums or the offered QoS and the generalization of the

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Figure 10: Performance gains when each provider models users with 9 clusters and offers a differentnumber of dataplans compared to macroscopic modeling when w j

R, h j, and n j are correlated (top) andwhen they are independent (bottom), respectively.

framework for a larger number of providers and dataplans was difficult. Our following work pro-vided an analytical methodology for computing the Nash equilibriums (NEs) of users and providersin a wireless oligopoly at the macroscopic level [9]. Unfortunately its extension at higher levels ofdetail was challenging due to various technical difficulties.

As mentioned in the introduction, this work extends our earlier papers and the state-of-the-art inthe following ways: (1) It enables providers to model users at different levels of detail and analyticallycomputes the equilibriums of users and providers. (2) It employs a detailed model of the network. (3)It allows providers to offer several dataplans depending on user traffic demand and models the user-decision making, capturing important aspects of customer behaviour. (4) It relaxes the assumptionabout the user rationality in selecting providers and dataplans.

5 Conclusions and future workThe proposed multi-layer two-stage game-theoretical modelling framework for wireless markets dis-tinguishes different customer groups and models their decision making process. It also modelsproviders with different capacities in their wireless infrastructure and degrees of knowledge aboutthe customer population. The providers can offer multiple dataplans. To capture the user-decisionmaking more realistically, the framework also models the “stickiness” to a provider.

Under a macroscopic view of the market, providers make suboptimal decisions. When providersmodel the customer population in a higher detail, they can improve their revenue. Gains also exist

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when providers offer a larger number of dataplans. In the case that the customer willingness topay and tolerance on low data rate are independent, if providers share the same level of knowledgeabout users, their competition may become more severe, resulting in revenue loss. Additionally, theincrease of the number of dataplans may also result in revenue losses due to the similar willingness topay of heavy and light users in such scenario. When a provider models users at a high level of detail,its revenue strongly depends on the knowledge of the other providers about users. For example, if theother providers also model the users in a high degree of detail, the competition may get enhanced,resulting to revenue loss. On the other hand, when a provider models the customers at higher level ofdetail compared to the others, it can achieve significant benefits. The number of disconnected usersalso decreases.

The analysis shows how providers develop their strategies and how they focus on specific cus-tomer segments. For example, often the strong provider focuses on business-type of customers,while providers with less resources attract low-profile customers. However when the capacity of thenetwork is reached, the strong provider may develop a different strategy to attract users with lowwillingness to pay. Moreover offering a larger number of dataplans allows providers to charge thevarious customer segments more efficiently achieving a higher revenue, under the condition that thewillingness-to-pay, tolerance in data rate and traffic are not independent.

Service providers can apply such tools to assess the evolution of a market, for different customerprofile densities, dataplans, and traffic demand. The number of providers and their capacity mayalso change to evaluate their impact on the market dynamics. To address the congestion, cost re-duction, and revenue generation, operators currently are also looking for attracting allies within theInternet market through sponsored data, offloading, infrastructure sharing, MVNOs, and networkslicing. For example, in network slicing, an infrastructure provider offers network resources, serviceproviders buy such resources according to the expected customer demand, and customers select thebest plan/service provider. The proposed framework can be extended to analyze the dynamics andevolution of such new markets.

References[1] J. Antoniou and A. Pitsillides. 2007. 4G converged environment: Modeling network selection

as a game. In IEEE Mobile and Wireless Communications Summit.

[2] Sumit Banerjee, Tom Loozen, Matt Fanno, and Abhinav Saksena. 2017. The Future Communi-cations Service Provider: A blueprint for relevance in the converged Digital world. Accenture(2017).

[3] Gunter Bolch, Stefan Greiner, Hermann de Meer, and Kishor S. Trivedi. 1998. Queueing Net-works and Markov Chains: Modeling and Performance Evaluation with Computer Science Ap-plications. Wiley-Interscience, New York, NY, USA.

[4] S. Boyd and L. Vandenberghe. 2004. Convex optimization. Cambridge university press.

[5] Francesco Capozzi, Giuseppe Piro, Luigi Alfredo Grieco, Gennaro Boggia, and Pietro Camarda.2013. Downlink packet scheduling in LTE cellular networks: Key design issues and a survey.IEEE Communications Surveys & Tutorials 15, 2 (2013), 678–700.

22

[6] Cisco. 2017. Cisco Visual networking Index: Global Mobile Data Traffice Fore-cast Update 2016-2021. White Paper (2017). https://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/mobile-white-paper-c11-520862.html

[7] Yong Cui, Tianze Ma, and Xiuzhen Cheng. 2011. Multi-hop access pricing in public areaWLANs. In Proc. IEEE Int. Conf. Comput. Commun.

[8] Abdolreza Eshghi, Dominique Haughton, and Heikki Topi. 2007. Determinants of customerloyalty in the wireless telecommunications industry. Telecommunications policy 31, 2 (2007),93–106.

[9] G. Fortetsanakis, I. Dimitriou, and M. Papadopouli. 2017. A game-Theoretical Analysis ofWireless Markets Using Network Aggregation. IEEE Trans. on Mobile Comput. 16, 3 (Mar.2017), 602–616.

[10] G. Fortetsanakis and M. Papadopouli. 2015. On Multi-Layer Modeling and Analysis of WirelessAccess Markets. IEEE Trans. Mobile Comput. 14, 1 (Jan. 2015), 113–125.

[11] Vojislav Gajic, Jianwei Huang, and Bixio Rimoldi. 2014. Competition of wireless providers foratomic users. IEEE/ACM Trans. Netw. 22, 2 (2014), 512–525.

[12] Vojislav Gajic, Jianwei Huang, and Bixio Rimoldi. 2014. Competition of wireless providers foratomic users. IEEE/ACM Transactions on Networking (TON) 22, 2 (2014), 512–525.

[13] Lin Gao, G. Iosifidis, Jianwei Huang, L. Tassiulas, and Duozhe Li. 2014. Bargaining-BasedMobile Data Offloading. IEEE J. Sel. Areas Commun. 32, 6 (2014).

[14] Lin Gao, Xinbing Wang, Youyun Xu, and Qian Zhang. 2011. Spectrum Trading in CognitiveRadio Networks: A Contract-Theoretic Modeling Approach. IEEE J. Sel. Areas Commun. 29,4 (Apr. 2011).

[15] S. H. Han, S. X. Lu, and S. C. Leung. 2012. Segmentation of telecom customers based oncustomer value by decision tree model. Expert Systems with Applications 39, 4 (2012), 3964–3973.

[16] Juncheng Jia and Qian Zhang. 2008. Competitions and dynamics of duopoly wireless serviceproviders in dynamic spectrum market. In Proc. 9th ACM Int. Symp. Mobile Ad Hoc Netw.Comput.

[17] Juncheng Jia, Qian Zhang, Qin Zhang, and Mingyan Liu. 2009. Revenue generation for truthfulspectrum auction in dynamic spectrum access. In Proc. 10th ACM Int. Symp. Mobile Ad HocNetw. Comput. 3–12.

[18] Mehmet Karaçuka, A Nazif Çatık, and Justus Haucap. 2013. Consumer choice and local net-work effects in mobile telecommunications in Turkey. Telecommunications Policy 37, 4 (2013),334–344.

[19] Mohammad T Kawser, MAB Hasib, Abduhu R Hasin, Adil MJ Sadik, and Ibrahim K Razu.2012. Performance comparison between round robin andproportional fair scheduling methodsfor LTE. International Journal of Information and Electronics Engineering 2, 5 (2012).

23

https://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/mobile-white-paper-c11-520862.html



[20] Frank P Kelly, Aman K Maulloo, and David KH Tan. 1998. Rate control for communication net-works: shadow prices, proportional fairness and stability. Journal of the Operational Researchsociety 49, 3 (1998), 237–252.

[21] M. Y. Kiang, M. Y. Hu, and D. M. Fisher. 2006. An extended self-organizing map network formarket segmentation - a telecommunication example. Decision Support Systems 42, 1 (2006).

[22] M. K. Kim, M. C. Park, and D. H. Jeong. 2004. The effects of customer satisfaction andswitching barrier on customer loyalty in Korean mobile telecommunication services. Telecom-munications policy 28, 2 (2004).

[23] Leonard Kleinrock. 1975. Queueing Systems, Volume 1: Theory. Wiley.

[24] Dapeng Li, Youyun Xu, Xinbing Wang, and Mohsen Guizani. 2011. Coalitional game theoreticapproach for secondary spectrum access in cooperative cognitive radio networks. IEEE Trans.on Wireless Commun. 10, 3 (2011), 844–856.

[25] Hsin-Hui Lin and Yi-Shun Wang. 2006. An examination of the determinants of customer loyaltyin mobile commerce contexts. Information & management 43, 3 (2006), 271–282.

[26] Jia-Shi Lin and Kai-Ten Feng. 2014. Femtocell access strategies in heterogeneous networksusing a game theoretical framework. IEEE Trans. on Wireless Commun. 13, 3 (2014), 1208–1221.

[27] Dusit Niyato and Ekram Hossain. 2008. Competitive pricing for spectrum sharing in cognitiveradio networks: Dynamic game, inefficiency of nash equilibrium, and collusion. IEEE J. Sel.Areas Commun. 26, 1 (2008), 192–202.

[28] Dusit Niyato and Ekram Hossain. 2008. Competitive pricing in heterogeneous wireless accessnetworks: Issues and approaches. IEEE Network 22, 6 (2008), 4–11.

[29] D. Niyato and E. Hossain. 2010. A Microeconomic Model for Hierarchical Bandwidth Sharingin Dynamic Spectrum Access Networks. IEEE Trans. Comput. 59, 7 (Jul. 2010), 865 –877.

[30] Dusit Niyato, Ekram Hossain, and Zhu Han. 2009. Dynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radio networks: A game-theoretic modeling approach. IEEETrans. Mobile Comput. 8, 8 (2009), 1009–1022.

[31] Ramneek, Patrick Hosein, Wonjun Choi, and Woojin Seok. 2016. A study of QoS support,performance and pricing of mobile data plans in the USA and South Korea. ICACT (2016).

[32] Shaolei Ren, Jaeok Park, and Mihaela Van der Schaar. 2011. User subscription dynamics andrevenue maximization in communications markets. In Proc. IEEE Int. Conf. Comput. Commun.

[33] Luca Rose, E Veronica Belmega, Walid Saad, and Mérouane Debbah. 2014. Pricing in Hetero-geneous Wireless Networks: Hierarchical Games and Dynamics. IEEE Trans. Wireless Com-mun. 13, 9 (2014), 4985–5001.

[34] RadhikaRanjan Roy. 2011. Fluid-Flow Mobility. In Handbook of Mobile Ad Hoc Networks forMobility Models. Springer US, 405–441.

24

[35] William H. Sandholm. 2015. Population Games and Deterministic Evolutionary Dynamics.Handbook of Game Theory with Economic Applications 4 (2015), 703–778.

[36] Amit M Schejter, Alexander Serenko, Ofir Turel, and Mehdi Zahaf. 2010. Policy implicationsof market segmentation as a determinant of fixed-mobile service substitution: What it meansfor carriers and policy makers. Telematics and Informatics 27, 1 (2010), 90–102.

[37] DongBack Seo, C Ranganathan, and Yair Babad. 2008. Two-level model of customer retentionin the US mobile telecommunications service market. Telecommunications Policy 32, 3 (2008),182–196.

[38] Gunnvald B Svendsen and Nina K Prebensen. 2013. The effect of brand on churn in the telecom-munications sector. European Journal of Marketing 47, 8 (2013), 1177–1189.

[39] Gerald Tellis and Gary Gaeth. 1990. Best value, price-seeking, and price aversion: the impactof information and learning on consumer choices. Journal of Marketing (April 1990).

[40] R. Trestian, O. Ormond, and G. M. Muntean. 2012. Game theory-based network selection:solutions and challenges. Communications Surveys & Tutorials 14, 4 (Jan. 2012), 1212–1231.

[41] T. Wysocki and A. Jamalipour. 2010. Pricing of Cognitive Radio Rights to Maintain the Risk-Reward of Primary User Spectrum Investment. In Proc. IEEE Symp. New Frontiers DynamicSpectrum Access Networks. 1 –8.

[42] P. Xavier and D. Ypsilanti. 2008. Switching costs and consumer behaviour: implications fortelecommunications regulation. info 10, 4 (2008).

[43] Xu Chen Junshan Zhang Xiaowen Gong, Lingjie Duan. 2017. When Social Network EffectMeets Congestion Effect in Wireless Networks: Data Usage Equilibrium and Optimal Pricing.IEEE Journal on Selected Areas in Communications (January 2017), 449–462.

[44] Lei Yang, Hongseok Kim, Junshan Zhang, Mung Chiang, and Chee Wei Tan. 2011. Pricing-based spectrum access control in cognitive radio networks with random access. In Proc. IEEEInt. Conf. Comput. Commun. 2228–2236.

[45] L. Ye, C. Qiuru, X. Haixu, L. Yijun, and Z. Guangping. 2013. Customer segmentation fortelecom with the k-means clustering method. Information Technology Journal 12, 3 (2013).

[46] Hui Yu, Lin Gao, Zheng Li, Xinbing Wang, and E. Hossain. 2010. Pricing for Uplink PowerControl in Cognitive Radio Networks. IEEE Trans. Veh. Technol. 59, 4 (May 2010), 1769–1778.

[47] Mengyuan Zhang, Lei Yang, and Xiaowen Gong. 2018. Wireless Service Pricing Competitionunder Network Effect, Congestion Effect, and Bounded Rationality. IEEE Trans. VehicularTechnology (April 2018).

[48] Xuejun Zhuo, Wei Gao, Guohong Cao, and Sha Hua. 2014. An Incentive Framework for Cel-lular Traffic Offloading. IEEE Transactions on Mobile Computing 13, 3 (2014).

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